Intersection property and interaction decomposition

The decomposition into interaction subspaces is a hierarchical decomposition of the spaces of cylindrical functions of a finite product space, also called factor spaces. It is an important construction in graphical models and a standard way to prove the Hammersley-Clifford theorem that relates Markov fields to Gibbs fields and plays a central role in Kellerer's result for the linearized marginal problem. We define an intersection of sum property, or simply intersection property, and show that it characterizes collections of vector subspaces over a poset that can be hierarchically decomposed into direct sums, giving therefore a general setting for such construction to hold. We will call this generalization the interaction decomposition. The intersection property is the Bayesian intersection property when specified to factor spaces which, under this new perspective on the interaction decomposition, appears to be a structure property. An application is the extension of the decomposition into interaction subspaces for any product of any set.